where \(l\)  represents the time instant, and \({\mathbf {\Lambda }} ({\mathbf {q}}_{\text{s}}(l)) = \left\lbrace \min \left\lbrace  Q_1, \Delta _1 ({\mathbf {q}}_{\text{s}}(l))\right\rbrace , \cdots , \min \left\lbrace  Q_N, \Delta _N ({\mathbf {q}}_{\text{s}}(l))\right\rbrace \right\rbrace \) . We denote this data power control algorithm for cellular links by DPCC. By proving that \({\mathbf {\Lambda }} ({\mathbf {q}}_{\text{s}})\)  is standard [1]}, we can readily verify that the DPCC algorithm converges to the optimal solution for any initial power vector \({\mathbf {q}}_{\text{s}} \succeq {\mathbf {0}}\) . The detailed proof is given in Appendix D.
